Q: prove that the following equation has no solution. (√(x +⌊ x ⌋)) + (√(x −(√x) )) = 1


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Question Number 175653 by mnjuly1970 last updated on 04/Sep/22

$Q : p r o v e t h a t t h e f o l l o w i n g$ $e q u a t i o n h a s n o s o l u t i o n .$ $\sqrt{x + ⌊ x ⌋} + \sqrt{x - \sqrt{x}} = 1$
	Answered by mahdipoor last updated on 04/Sep/22

	$f (x) = \sqrt{x - \sqrt{x}} + \sqrt{x + [x]}$ $D_{f} = \forall x ⩾ 1$ $f o r I_{n} = [n, n + 1) n ⩾ 1$ $f_{n} (x) = \sqrt{x - \sqrt{x}} + \sqrt{x + n}$ $D_{x} (f_{n} (x)) = \frac{1 - \frac{1}{2 \sqrt{x}}}{2 \sqrt{x - \sqrt{x}}} + \frac{1}{2 \sqrt{x + n}} > 0$ $f i s i n c r i s e f u n c t i o n i n I_{n} \Rightarrow$ $R_{f} i n I_{n} = [f_{n} (n), f_{n} (n + 1))$ $\Rightarrow\Rightarrow$ $R_{f} i n D_{f} = I_{1} \cap I_{2} \cap . . . =$ $[\sqrt{2}, \sqrt{3} + \sqrt{2 - \sqrt{2}}) \cap [\sqrt{4} + \sqrt{2 - \sqrt{2}}, \sqrt{5} + \sqrt{3 - \sqrt{3}}) \cap . . .$ $\Rightarrow \sqrt{2} ⩽ R_{f}$ $\Rightarrow f (x) \neq 1 f o r \forall x \in D_{f}$
	Commented by mnjuly1970 last updated on 04/Sep/22

	$t h a n k s a l o t s i r m a h d i p o o r$
	Answered by mr W last updated on 04/Sep/22

	$x ⩾ 0$ $x ⩾ \sqrt{x} \Rightarrow x ⩾ 1 \Rightarrow ⌊ x ⌋ ⩾ 1$ $x + ⌊ x ⌋ ⩾ 2$ $\sqrt{x + ⌊ x ⌋} ⩾ \sqrt{2}$ $\sqrt{x + ⌊ x ⌋} + \sqrt{x - \sqrt{x}} ⩾ \sqrt{2} > 1$ $\sqrt{x + ⌊ x ⌋} + \sqrt{x - \sqrt{x}} = 1 c a n n e v e r b e t r u e .$ $i . e . \sqrt{x + ⌊ x ⌋} + \sqrt{x - \sqrt{x}} = 1 h a s n o s o l u t i o n .$
	Commented by mnjuly1970 last updated on 04/Sep/22

	$b r a v o s i r W$
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